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Math225_M1_sol

# Math225_M1_sol - Midterm 1 Solutions 1(a Find 2 2 matrices...

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Midterm 1 Solutions 1. (a) Find 2 × 2 matrices A and B such that AB 6 = BA . There are many examples to answer this problem. Here is one: Let A = 1 0 0 0 and B = 0 1 1 1 . Then AB = 0 1 0 0 and BA = 0 0 1 0 so that AB 6 = BA . (b) For general n × n matrices A and B find an expression for ( A + B ) 2 . Let A and B be n × n matrices. Then, ( A + B ) 2 = ( A + B )( A + B ) = A ( A + B ) + B ( A + B ) = A 2 + AB + BA + B 2 . Where the last two equalities are satisfied by distributive properties of matrix operations. (c) Write the following system of Differential Equations in matrix vector form: x 0 1 = 3 x 1 + 5 x 2 + sin( t ) , x 0 2 = x 1 + 2 x 2 + cos( t ) x 0 1 x 0 2 = 3 5 1 2 x 1 x 2 + sin( t ) cos( t ) 2. Answers without correct supporting work will not count, even if correct. Let A = 1 - 1 2 2 1 11 4 - 3 10 . Compute A - 1 using Elementary Row Transformation and check your work by computing A - 1 A . 1 - 1 2 | 1 0 0 2 1 11 | 0 1 0 4 - 3 10 | 0 0 1 1 1 - 1 2 | 1 0 0 0 3 7 | - 2 1 0 0 1 2 | - 4 0 1 2 1 - 1 2 | 1 0 0 0 1 2 | - 4 0 1 0 3 7 | - 2 1 0 3 1 0 4 | - 3 0 1 0 1 2 | - 4 0 1 0 0 1 | 10 1 - 3 4 1 0 0 | - 43 - 4 13 0 1 0 | - 24 - 2 7 0 0 1 | 10 1 - 3 .

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Math225_M1_sol - Midterm 1 Solutions 1(a Find 2 2 matrices...

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